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Efficient and Numerically Stable Sparse Learning Sihong Xie1, Wei Fan2, Olivier Verscheure2, and Jiangtao Ren3 1University of Illinois at Chicago, USA 2 IBM T.J. Watson Research Center, New York, USA 3 Sun Yat-Sen University, Guangzhou, China Applications Signal processing (compressive sensing, MRI, coding, etc.) Computational Biology (DNA array sensing, gene expression pattern annotation ) Geophysical Data Analysis Machine learning Algorithms Greedy selection Via L-0 regularization Boosting, forward feature selection not for large scale problem Convex optimization Via L-1 regularization (e.g. Lasso) IPM (interior point method) medium size problem Homotopy method full regularization path computation Gradient descent Online algorithm (Stochastic Gradient Descent) Rising awareness of Numerical Problems in ML Efficiency SVM, beyond Optimization black box solver Large scale problems, parallelization Eigenvalue problems, randomization Stability Gaussian process calculation, solving large system of linear equations, matrix inversion Convergence of gradient descent, matrix iteration computation For more topics of numerical mathematics in ML, See : ICML Workshop on Numerical Methods in Machine Learning 2009 Stability in Sparse learning Iterative Hard Thresholding (IHT) Solve the following optimization problem Incorporating gradient descent with hard thresholding Stability in Sparse learning Iterative Hard Thresholding (IHT) Simple and scalable With RIP assumption, previous methods [BDIHT09, GK09] shows that iterative hard thresholding converges. Without the assumption of the spectral radius of the iteration matrix, such methods may diverge. Stability in Sparse learning Gradient Descent with Matrix Iteration Stability in Sparse learning Mirror Descent Algorithm for Sparse Learning (SMIDAS) Stability in Sparse learning Elements of the Primal Vector is exponentially sensitive to the corresponding elements of the Dual Vector Stability in Sparse learning Example Suppo
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